Reciprocity, transitivity, and the mysterious three-cycle

Abstract Reciprocity and transitivity are the two most important structural mechanisms underlying friendship network evolution. While on their own they are understood in great detail, the relation between them is rarely studied systematically. Are friendships outside of social groups more or less likely to be reciprocated than friendships embedded in a group? Using a theoretical framework that focusses on the situations in which friends interact and the social structures that stabilise one-sided friendships, I propose that the tendency towards reciprocation of friendships within transitive groups is usually lower than outside of transitive groups. In a meta-analysis of two datasets including 29 friendship networks using stochastic actor-oriented models (SAOMs), the interaction between reciprocity and transitivity is analysed. Supporting the theoretical reasoning, the interaction is consistently negative. Second, the tendency against forming three-cycles in friendship networks, which was consistently found in previous studies, is shown to be spurious and a result of neglecting to control for the tendency against reciprocation in transitive groups. The tendency against three-cycles is commonly seen as an indicator that unreciprocated friendships indicate local hierarchy differences between individuals; this proposition has to be re-evaluated in light of the findings of this study. Future studies that analyse the evolution of friendship networks should consider modelling reciprocation in transitive triplets and potentially omit modelling three-cycles.